2018 A/L Physics Past Paper · Question 32 (English Medium)

Answer: (2) 92, 146

Given information

The last nucleus is stable $^{206}_{82}\text{Pb}$.

On the way, X gives out $8\,\alpha$ and $6\,\beta^-$.

The first nucleus is $^{A}_{Z}X$.

What the question asks

We want the number of protons $Z$ and the number of neutrons $A - Z$ in X.

Relevant theory

This equation states that the nucleus X having the mass number A and the atomic number Z becomes the nucleus Y which has the mass number A-4 and the atomic number Z-2 after emitting a helium nucleus.

In this process, the mass number remains unchanged and the atomic number increases by one.

In every decay, two totals stay the same.

The total mass number (the number of nucleons) stays the same.

The total charge (the atomic number) stays the same.

An $\alpha$ particle ($^4_2\text{He}$) takes away 4 nucleons and a charge of $+2e$.

In $\beta^-$ decay a neutron turns into a proton. The number of nucleons stays the same. The atomic number rises by 1.

Required equations

$A = 206 + 8 \times 4 + 6 \times 0$

$Z = 82 + 8 \times 2 - 6 \times 1$

Applying it to this question

At the start: $^{A}_{Z}X$.

At the end: $^{206}_{82}\text{Pb}$, $8\,\alpha$ and $6\,\beta^-$.

$238$

$206 + 8 \times 4 + 6 \times 0$

$92$

$82 + 8 \times 2 + 6 \times (-1)$

Reasoning and calculation

$238$

$A = 206 + 8 \times 4$

$92$

$Z = 82 + 8 \times 2 - 6$

$Z$

$146$

$N = A - Z = 238 - 92$

Checks

$206 + 8 * 4 - (82 + 8 * 2 - 6)$

Check: $^{238}_{92}\text{U}$ to $^{206}_{82}\text{Pb}$.

The mass number falls by $32 = 8 \times 4$. All of it is from the $\alpha$ particles.

The atomic number falls by $10 = 8 \times 2 - 6$.

Both totals balance. $Z = 92$ is uranium.

Conclusion

$^{238}_{92}\text{U}$

X has 92 protons and 146 neutrons. It is $^{238}_{92}\text{U}$.

The answer is option (2).

Taking a $\beta^-$ decay to lower the atomic number.

In $\beta^-$ decay a neutron turns into a proton. The atomic number rises by 1.

So when you work back to X, take the 6 away.

Explore

A nucleus X is not stable. It starts to break down.

It throws out an $\alpha$ particle: 2 protons and 2 neutrons. $A$ falls by four and $Z$ falls by two. This happens eight times.

Six times, a neutron inside turns into a proton and an electron flies out. That is a $\beta^-$ particle. $Z$ rises by one each time.

In the end it is stable lead, $^{206}_{82}\text{Pb}$.

Go back: $A = 206 + 32 = 238$ and $Z = 82 + 16 - 6 = 92$. So X has 92 protons and 146 neutrons: option (2).